A Review on Nano Fluids - Part i Theoretical and Numerical Investigations

8/8/2019 A Review on Nano Fluids - Part i Theoretical and Numerical Investigations

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ISSN 0104-6632Printed in Brazil

www.abeq.org.br/bjche

Vol. 25, No. 04, pp. 613 - 630, October - December, 2008

*To whom correspondence should be addressed

Brazilian Journal

of Chemical

Engineering

A REVIEW ON NANOFLUIDS - PART I:

THEORETICAL AND NUMERICAL

INVESTIGATIONS

Xiang-Qi Wang and Arun S. Mujumdar1

Department of Mechanical Engineering, National University of Singapore,

Phone: +(65) 6874-4623, Fax: +(65) 6779-1459,

10 Kent Ridge Crescent, 119260, Singapore.E-mail: [email protected]

(Received: January 2, 2008 ; Accepted: May 20, 2008)

Abstract - Research in convective heat transfer using suspensions of nanometer-sized solid particles in base

liquids started only over the past decade. Recent investigations on nanofluids, as such suspensions are often

called, indicate that the suspended nanoparticles markedly change the transport properties and heat transfer

characteristics of the suspension. This first part of the review summarizes recent research on theoretical and

numerical investigations of various thermal properties and applications of nanofluids.

Keywords: Nanofluids; Nanoparticles; Heat transfer; Thermal conductivity.

INTRODUCTION

Convective heat transfer can be enhanced

passively by changing flow geometry, boundary

conditions, or by enhancing thermal conductivity of

the fluid. Various techniques have been proposed to

enhance the heat transfer performance of fluids.

Researchers have also tried to increase the thermal

conductivity of base fluids by suspending micro- or

larger-sized solid particles in fluids, since the

thermal conductivity of solid is typically higher than

that of liquids, as seen from Table 1. Numeroustheoretical and experimental studies of suspensions

containing solid particles have been conducted since

Maxwells theoretical work was published more than

100 years ago (Maxwell, 1881). However, due to the

large size and high density of the particles, there is no

good way to prevent the solid particles from settling

out of suspension. The lack of stability of such

suspensions induces additional flow resistance and

possible erosion. Hence, fluids with dispersed coarse-

grained particles have not yet been commercialized.

Table 1: Thermal conductivities of various solids and liquids

Material Thermal Conductivity (W/m-K)

copper 401Metallic solids

aluminum 237

silicon 148Nonmetallic solids

alumina (Al2O3) 40

Metallic liquids sodium (644 K) 72.3

water 0.613

ethylene glycol (EG) 0.253Nonmetallic liquids

engine oil (EO) 0.145


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614 Xiang-Qi Wang and Arun S. Mujumdar

Brazilian Journal of Chemical Engineering

Modern nanotechnology provides new

opportunities to process and produce materials with

average crystallite sizes below 50 nm. Fluids with

nanoparticles suspended in them are called

nanofluids, a term proposed in 1995 by Choi of the

Argonne National Laboratory, U.S.A. (Choi, 1995).

Nanofluids can be considered to be the next-generation heat transfer fluids because they offer

exciting new possibilities to enhance heat transfer

performance compared to pure liquids. They are

expected to have superior properties compared to

conventional heat transfer fluids, as well as fluids

containing micro-sized metallic particles. The much

larger relative surface area of nanoparticles,

compared to those of conventional particles, should

not only significantly improve heat transfer

capabilities, but also should increase the stability of

the suspensions. Also, nanofluids can improve

abrasion-related properties as compared to the

conventional solid/fluid mixtures. Successfulemployment of nanofluids will support the current

trend toward component miniaturization by enabling

the design of smaller and lighter heat exchanger

systems. Keblinski et al. (2005), in an interesting

review, discussed the properties of nanofluids and

future challenges. The development of nanofluids is

still hindered by several factors such as the lack of

agreement between results, poor characterization of

suspensions, and the lack of theoretical

understanding of the mechanisms.

Suspended nanoparticles in various base fluids

can alter the fluid flow and heat transfer

characteristics of the base fluids. Necessary studiesneed to be carried out before wide application can be

found for nanofluids. In this paper we present an

overview of the literature dealing with recent

developments in the study of heat transfer using

nanofluids. First, the preparation of nanofluids is

discussed; this is followed by a review of recent

experimental and analytical investigations with

nanofluids.

THEORETICAL INVESTIGATIONS

Mechanisms of Nanofluids

The conventional understanding of the effective

thermal conductivity of mixtures originates from

continuum formulations which typically involve only

the particle size/shape and volume fraction and

assume diffusive heat transfer in both fluid and solid

phases. This method can give a good prediction for

micrometer or lager-size solid/fluid systems, but it

fails to explain the unusual heat transfer

characteristics of nanofluids.

The present understanding of thermal transport in

nanofluids can be grouped into two categories. Some

postulate that the thermal conductivity of nanofluids

is composed of the particles conventional static partand a Brownian motion part which produces micro-

mixing. These models take the particle dynamics into

consideration, whose effect is additive to the thermal

conductivity of a static dilute suspension. Thus, the

particle size, volume fraction, thermal conductivities

of both the nanoparticle and the base fluid, and the

temperature itself are taken into account in such

models for the thermal conductivity of nanofluids.

These theories provide a means of understanding the

particle interaction mechanism in nanofluids.

Other groups have started from the nanostructure

of nanofluids. These investigators assume that the

nanofluid is a composite, formed by the nanoparticleas a core, and surrounded by a nanolayer as a shell,

which in turn is immersed in the base fluid, and from

which a three-component medium theory for a

multiphase system is developed. Some have

suggested that the enhancement is due to the ordered

layering of liquid molecules near the solid particles.

The mechanism for thermal conduction between a

liquid and a solid is not clear.

To explain the reasons for the anomalous increase

of the thermal conductivity in nanofluids, Keblinski

et al. (2002) and Eastman et al. (2004) proposed

four possible mechanisms, e.g., Brownian motion of

the nanoparticles, molecular-level layering of theliquid at the liquid/particle interface, the nature of

heat transport in the nanoparticles, and the effects of

nanoparticle clustering, which are schematically

shown in Fig. 1. They postulated that the effect of

Brownian motion can be ignored, since the

contribution of thermal diffusion is much greater

than Brownian diffusion. However, they only

examined the cases of stationary nanofluids. Wang

et al. (1999) argued that the thermal conductivities of

nanofluids should be dependent on the microscopic

motion (Brownian motion and inter-particle forces)

and particle structure. Xuan and Li (2000) also

discussed four possible reasons for the improved

effective thermal conductivity of nanofluids: the

increased surface area due to suspended

nanoparticles, the increased thermal conductivity of

the fluid, the interaction and collision among

particles, the intensified mixing fluctuation and

turbulence of the fluid, and the dispersion of

nanoparticles.


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A Review on Nanofluids - Part I Theoretical and Numerical Investigations 615

Brazilian Journal of Chemical Engineering Vol. 25, No. 04, pp. 613 - 630, October - December, 2008

0 5 10 15 20 25 30

d (nm)

1

1.5

2

2.5

3

d d+2h

0.5 nm1 nm

1.5 nm2 nm

k

0 5 10 15 20 25 30

d (nm)

1

1.5

2

2.5

3

d d+2h

0.5 nm1 nm

1.5 nm2 nm

0.5 nm1 nm

1.5 nm2 nm

k

(a)

(b)

iv

iii

ii

i

0.40.20 0.6 0.8 1

15

12

9

6

3

k

iv

iii

ii

i

0.40.20 0.6 0.8 1

15

12

9

6

3

k

(c)

Figure 1: Schematic diagrams of several possible mechanisms (Keblinski et al., 2002): (a) Enhancement

of k due to formation of highly conductive layer-liquid structure at the liquid/particle interface;

(b) Ballistic and diffusive phonon transport in a solid particle; (c)Enhancement

of k due to increased effective of highly conducting clustersMany researchers used the concept of liquid/solid

interfacial layer to explain the anomalous improvement

of the thermal conductivity in nanofluids. Yu and

Choi (2003, 2004) suggested models, based on

conventional theory, which consider a liquid molecularlayer around the nanoparticles. However, a study of

Xue et al. (2004) using molecular dynamics simulationshowed that simple monatomic liquids had no effect on

the heat transfer characteristics both normal and

parallel to the surface. This means that thermal

transport in a layered liquid may not be adequate to

explain the increased thermal conductivity ofsuspensions of nanoparticles.

Khaled and Vafai (2005) investigated the effect

of thermal dispersion on heat transfer enhancement

of nanofluids. These results showed that the presenceof the dispersive elements in the core region did not

affect the heat transfer rate. However, thecorresponding dispersive elements resulted in 21%

improvement of the Nusselt number for a uniform

tube supplied by a fixed heat flux as compared to theuniform distribution for the dispersive elements.

These results provide a possible explanation for the

increased thermal conductivity of nanofluids, which

may be determined partially by the dispersiveproperties.

Wen and Ding (Wen and Ding, 2005; Ding and

Wen, 2005) studied theoretically the effect of

particle migration on heat transfer characteristics innanofluids flowing through mini-channels

( D 1= mm). They studied the effect of shear-induced and viscosity-gradient-induced particlemigration and the self-diffusion due to Brownian

motion. Their results indicated a significant non-

uniformity in particle concentration and thermal

conductivity over the tube cross-section due toparticle migration. Compared to the uniform

distribution of thermal conductivity, the non-uniform distribution caused by particle migration

induced a higher Nusselt number.


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Koo and Kleinstreuer (2005a) discussed the effects

of Brownian, thermo-phoretic, and osmo-phoreticmotions on the effective thermal conductivities. They

found that the role of Brownian motion is much more

important than the thermo-phoretic and osmo-phoreticmotions. Furthermore, the particle interaction can be

neglected when the nanofluid concentration is low ( 100), can be expressed as follows:

( ) ( ) ( )( ) ( )

p b b p

eff b

p b b p

k n 1 k n 1 k k k k

k n 1 k k k

+ =

+ + (7)

where n is the empirical shape factor given by

n 3/= , and is the particle sphericity, defined asthe ratio of the surface area of a sphere with volume

equal to that of the particle, to the surface area of theparticle. Comparison between Eq. (7) and Eq. (5)

reveals that Maxwells model is a special case ofHamilton and Crossers model for sphericity equal to

one.

As discussed earlier, the classical models are

derived from continuum formulations and includeonly the particle shape and volume fraction as

variables and assumed diffusive heat transport inboth liquid and solid phases. The large enhancement

of the effective thermal conductivity in nanofluids

defies Maxwells theory (Maxwell, 1881) as well asits modification by Hamilton and Crosser (1962).

Some important mechanisms in nanofluids appear to

be neglected in these models. Keblinski et al. (2002)

investigated the possible factors of enhancingthermal conductivity in nanofluids such as the size,

the clustering of particles, and the nano-layerbetween the nanoparticles and base fluids. Based on

the traditional models, many later theoretical works

have been proposed to address such effects,

especially the interfacial characteristics.Yu and Choi (2003) proposed a modified

Maxwell model to account for the effect of the nano-

layer by replacing the thermal conductivity of solidparticles pk in Eq. (5) with the modified thermal

conductivity of particles pek , which is based on the

so-called effective medium theory (Schwartz

et al., 1995):

( ) ( ) ( )

( ) ( ) ( )

3

pe p3

2 1 1 1 2k k

1 1 1 2

+ + + =

+ + + (8)

where layer pk k = is the ratio of nano-layer

thermal conductivity to particle thermal conductivityand h r = is the ratio of the nano-layer thicknessto the original particle radius. Hence, the Maxwell

equation (Eq. (5)) can be re-cast as follows:

( )( )

( )( )

3pe b pe b

eff b3pe b pe b

k 2k 2 k k 1k k

k 2k k k 1

+ + =

+ + (9)


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The new model including the nano-layer can

predict the presence of very thin nano-layers havinga thickness less than 10 nm. It also indicates that the

addition of smaller ( b > c) with surrounding nano-layers.

With a general empirical shape factor n ( n 3 = ,here is an empirical parameter and is theparticle sphericity), this modified HC model canpredict the thermal conductivity of carbon nanotube-

in-oil nanofluids reasonably well. However, it fails

to predict the nonlinear behavior of the effectivethermal conductivity of general oxide and metal-

based nanofluids.Xue (2003) developed a model for the effective

thermal conductivity of nanofluids. His model is

based on the average polarization theory and

includes the effect of the interface between the solidparticles and the base fluid. His formula for the

effective thermal conductivity is

( )

( )( )

eff b

eff b

eff c,x

eff 2,x c,x eff

eff c,y

eff 2,x c,y eff

k k9 1

2k k

k k

k B k k 0

k k4

2k 1 B k k

+ +

+

+ =

+

(11)

where = abc/[(a + t)(b + t)(c + t)] with half-radii(a,b,c) of the assumed elliptical complex

nanoparticles, which consist of nanoparticles and

interfacial shells between particles and the base

fluids. c, jk is the effective dielectric constant and

2,xB is the depolarization factor along the x-

symmetrical axis, which is derived from the average

polarization theory. A test of this formula (Kim et al.,

2004) reveals that it is not as accurate as Xueclaimed since he used incorrect values of theparameters such as the depolarization factor. Xue

and Xu (2005) obtained an equation for the effectivethermal conductivity according to Bruggeman`s

model (Bruggeman, 1935). Their model takes into

account the effect of interfacial shells by replacingthe thermal conductivity of nanoparticles with the

assumed thermal conductivity of the so-called

complex nanoparticles, which included theinterfacial shells between the nanoparticles and the

base fluids.

( )( ) ( ) ( )( )( ) ( )( )

eff b

eff b

eff 2 2 1 1 2 2 eff

eff 2 2 1 1 2 2 eff

k k12k k

k k 2k k k k 2k k 0

2k k 2k k 2 k k k k

+ +

+ +=

+ + +

(12)

where is the volume ratio of sphericalnanoparticle and complex nanoparticle, and k1 and k2

are the thermal conductivity of the nanoparticle and

interfacial shell, respectively. The modified model is

in good agreement with the experimental data on the

effective thermal conductivity of CuO/water and

CuO/EG nanofluids (Lee et al., 1999).Xie et al. (2005) considered the interfacial nano-

layer with linear thermal conductivity distribution

and proposed an effective thermal conductivitymodel to account for the effects of nano-layer

thickness, nanoparticle size, volume fraction, andthermal conductivities of fluid, nanoparticles, and

nano-layer. Their formula is

2 2T

eff T bT

3k 1 3 k

1

= + +

(13)

with

( ){ } ( )3 3lb pl bl lb pl1 1 2 = + + + ,

where

( ) ( )lb l b l bk k k 2k = + ,


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( ) ( )pl p l p lk k k 2k = + ,

( ) ( )bl b l b lk k k 2k = + ,

and pr = is the thickness ratio of nano-layer and

nanoparticle. T is the modified total volume fractionof the original nanoparticle and nano-layer,

( )3

T 1 = + . They claimed that the calculated values

agreed well with some available experimental data.

For metallic particles, Patel et al. (2003) found

9% enhancement of thermal conductivity even at

extremely low concentrations such as 0.00026%.

(Note that if not mentioned in the text, theconcentration is in volume fraction.) The previous

formulas fail to predict such strange phenomena. The

Brownian motion of nanoparticles at the molecular

and nano-scale levels may be a key mechanismgoverning the thermal behavior of nanofluids. Also,

in recent experiments (Das et al., 2003b), we findthat the thermal conductivity of nanofluids depends

strongly on temperature. Hence, this important fact

should be considered in theoretical models.

Xuan et al. (2003) considered the random motionof suspended nanoparticles (Brownian motion) based

on the Maxwell model and proposed a modified

formula for the effective thermal conductivity as

follows:

( )( )

p b b p p p B

eff bcp b b p

k 2k 2 k k c k T

k k 2 3 rk 2k k k

+

= + + + (14)

where the Boltzmann constant is23

Bk 1.381 10= J/K; cr is the apparent radius of

clusters and depends on the fractal dimension of thecluster structure. Although this model incorporates

the effect of temperature on the conductivity

enhancement, the dependence is too weak ( 1/ 2T )and not in agreement with the experimental data ofDas et al. (2003b).

Based on the fractal theory (Mandelbrot, 1982),

which can describe well the disorder and stochasticprocess of clustering and polarization of

nanoparticles within the mesoscale limit, a fractal

model for predicting the effective thermalconductivity of nanofluid was proposed by Wang

et al. (2003), who developed a fractal model based

on the multi-component Maxwell model by

substituting the effective thermal conductivity of the

particle clusters, clk (r) , and the radius distribution

function, n(r), as follows:

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

0 cl cl beff b

0 b cl b

1 3 k r n r / k r 2k drk k

1 3 k r n r / k r 2k dr

+ + = + +

(15)

This model fit successfully the experimental datafor a 50 nm CuO particle suspension in deionized

water with 0.5% < .Kumar et al. (2004) proposed a comprehensive

model to account for the large enhancement of

thermal conductivity in nanofluids and its strong

temperature dependence, which was deduced from

the Stokes-Einstein formula. The thermalconductivity enhancement taking account of the

Brownian motion of particles can be expressed as:

( ) ( )bBeff b b2

b pp

r2k Tk k c k k 1 rd

= +

(16)

where c is a constant, is the dynamic viscosity of

the base fluid, and dp is the diameter of the particles.

However, the validity of the model has got to be

established; it may not be suitable for high

concentrations of particles.Bhattacharya et al. (2004) developed a technique

to compute the effective thermal conductivity of a

nanofluid using Brownian motion simulation. Theycombined the liquid conductivity and particle

conductivity as follows

( )eff p bk k 1 k = + (17)

where pk is replaced by the effective contribution of

the particles towards the overall thermal conductivity

of the system,n

p 2 j 0B

1k Q(0)Q( j t) t

k T V == . The

model showed a good agreement with the thermalconductivity of nanofluids.

Jang and Choi (2004b) devised a theoretical

model that involves four modes such as collision

between base fluid molecules b(k (1 )) , thermaldiffusion in nanoparticles in fluids pk , collision

between nanoparticles due to Brownian motion

(neglected), and thermal interaction of dynamic or

dancing nanoparticles with the base fluid

molecules ( Tfh ). The resulting expression for theeffective thermal conductivity of nanofluids is


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( )p

2beff b p b d

p

dk k 1 k 3C k Re Pr

d= + + (18)

wherep

2 2bd

p

kh Re Pr

d and p3d represent the

heat transfer coefficient for the flow pastnanoparticles and the thickness of the thermal

boundary layer, respectively. The advantage of themodel is to include the effects of concentration,

temperature, and particle size. However, the

Brownian effect was neglected, which may not besuitable since the high temperature-dependent

properties may be caused by the Brownian motion.On the other hand, Prasher (Putnam et al., 2006)

proposed that convection caused by Brownian

motion of the nanoparticles is primarily responsible

for the enhancement of the effective thermalconductivity of nanofluids. By introducing the

general correlation for heat transfer coefficient h, hemodified the Maxwell model by including the

convection of the liquid near the particles due to

Brownian movement:

( )( )

p b p bm 0.333eff b

p b p b

k 2k 2 k k k 1 ARe Pr k

k 2k k k

+ + = +

+

(19)

where m 0.333bh k / a(1 A Re Pr )= + and A and m areconstants. The Reynolds number can be written as:

b

p p

18k T1Re

v d=

Recently, Koo and Kleinstreuer (2004, 2005b)

developed a new model for nanofluids, whichincludes the effects of particle size, particle volume

fraction and temperature dependence as well as

properties of the base fluid and the particle subject to

Brownian motion. The resulting formula is

( )

( )

( )

p b p b

eff b

p b p b

4 Bp p

p

k 2k 2 k k k k

k 2k k k

k T5 10 c f T,

D

+ + = +

+

(20)

Note that the first part of Eq. (20) is obtaineddirectly from the Maxwell model while the second

part accounts for Brownian motion, which causes the

temperature dependence of the effective thermal

conductivity. The function f (T, ) can be assumed tovary continuously with the particle volume fraction,f (T, ) ( 6.04 0.4705)T (1722.3 134.63) = + + while is related to particle motion. Based on theinvestigation of pressure gradients, temperature

profiles and Nusselt numbers, Koo andKleinstreuer (2005b) also claimed that addition of 1-

4% CuO nanoparticles and high-Prandtl number basefluid such as ethylene glycol and oils could

significantly increase the heat transfer performance

of micro-heat sinks.

Considering the fact that carbon nanotubes

possess a large aspect ratio, their thermalconductivity is more difficult to predict. Nan

et al. (2003) generalized the Maxwell-Garnett

approximation and derived a simple formula

eff p bk 1 k 3k = + to predict the effective thermal

conductivity of carbon-nanotube-based composites.The results within Nans model (Nan et al., 2003)

agree well with the experimental observations

(Choi et al., 2001). However, this model does not

consider the thermal resistance across the carbonnanotube-fluid interface. Later, Nan et al. (2004)

modified their model and tried to describe the effect

of the interface thermal resistance. However, the

model still cannot explain the nonlinear phenomena

of the effective thermal conductivity of nanotubesuspensions with nanotube volume fractions.

Recently, to account for the geometric and physicalanisotropy simultaneously, Gao and Zhou (2005)

proposed a differential effective medium theorybased on Bruggemans model (Bruggeman, 1935)

to predict the effective thermal conductivity ofnanofluids. Although their model involves the

effect of aspect ratio of the nanotube, the size effectand temperature dependence were not included.

From the results, the prediction of the thermal

conductivity of nanotube-based suspensions is not

good.For carbon nanofibers, Yamada and Ota (1980)

proposed the unit-cell model for the effective

thermal conductivity of mixture, which is

( )( )

p b p b

eff b

p b p b

k /k K K 1 k /k k k

k / k K 1 k / k

+ =

+ + (21)

where, K is the shape factor and 0.2 p pK 2 (l / d )= for

the cylindrical particles, and pl and pd are the length

and diameter of the cylindrical particle.


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Recently, Xue (2005) proposed a Maxwell

theory based model of the effective thermal

conductivity of carbon nanotube (CNT) nanofluids toinclude the effect of large axial ratio and the space

distribution of the CNTs. As compared with theexisting experimental data (Choi et al., 2001), the

proposed model provided reasonable agreement withadjusted thermal conductivity of CNTs. With the

assumed distribution function jP(B ) 2= , the

corresponding expression for the effective thermal

conductivity of CNT-based nanofluids is

p p b

p b beff b

p bb

p b b

k k k1 2 ln

k k 2k k k

k kk1 2 ln

k k 2k

+ +

=

+ +

(22)

Xue (2006) also presented a model for theeffective thermal conductivity of carbon nanotube

composites by incorporating the interface thermal

resistance with an average polarization theory. The

proposed model includes the effects of carbon

nanotube length, diameter, concentration, interface

thermal resistance, and the thermal conductivities of

nanotube and base fluid on the thermal conductivity

of the nanofluid simultaneously.

( )

( ) ( )

eff b

eff b

c ceff 33 eff 11

c c

eff 33 eff eff 11 eff

k k9 1

2k k

k k k k 4 0

d 1

k 0.14 k k 2k k k L 2

+

+

+ =

+ +

(23)

where c11k andc33k are the transverse and

longitudinal equivalent thermal conductivities of the

composite unit cell of a nanotube with length L anddiameter d. The model predicts that the nanotube

diameter has a negligible effect on the thermal

conductivity enhancement of the nanotubenanofluids. It seems that the model agrees well with

the data of Xie et al. (2003).

Fig. 2 shows a comparison between selected

discussed theoretical models and experimental data

on thermal conductivity of Al2O3 /water nanofluids.To obtain a deeper understanding of the effective

thermal conductivity of nanofluids, some important

facts must be taken to account in future studies. Suchfacts include the effect of the size and shape of the

nanoparticles, the interfacial contact resistancebetween nanoparticles and base fluids, the

temperature dependence or the effect of Brownian

motion, and the effect of clustering of particles.

1.50

1.45

1.40

1.35

1.30

1.25

1.20

1.15

1.10

1.05

1.00

volume fraction

0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040 0.045 0.050 0.055 0.060

k_

eff/k_

b

1.50

1.45

1.40

1.35

1.30

1.25

1.20

1.15

1.10

1.05

1.00

volume fraction

0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040 0.045 0.050 0.055 0.060

k_

eff/k_

b

Figure 2: Comparison between selected theoretical models and experimental

data for thermal conductivity for Al2O3/water nanofluids.


10/18



Viscosity

Einstein (1956) was the first to calculate theeffective viscosity of a suspension of spherical solidsusing the phenomenological hydrodynamic equations.By assuming that the disturbance of the flow pattern

of the matrix base fluid caused by a given particledoes not overlap with the disturbance of flow causedby the presence of a second suspended particle, hederived the following equations

( )eff p b1 2.5 = + (24)

Even since Einsteins initial work, researchershave made progress in extending the Einstein theoryin three major areas.1. To extend the Einstein equation to higher particlevolume concentrations by including particle-particleinteractions. The theoretical equation can be

expressed as:2 3

eff 1 p 2 p 3 p b(1 c c c ) = + + + + .2. To take into account the fact that the effectiveviscosity of a mixture becomes infinite at the

maximum particle volume concentration p max .3.

This theoretical equation usually has the term

( )p p max1

in the denominator, which can be

expressed in a form similar to (1).To include the effect of non-spherical particle

concentrations. Some of these equations are included

in Table 2.Experimental data for the effective viscosity ofnanofluids are limited to certain nanofluids. Theranges of the parameters (the particle volumeconcentration, temperature, etc.) are also limited.Still, the experimental data show the trend that theeffective viscosities of nanofluids are higher than theexisting theoretical predictions. In an attempt torectify this situation, researchers proposed equationsapplied to specific applications, e.g., Al2O3 in water(Maiga et al., 2004a), Al2O3 in ethylene glycol(Maiga et al., 2004a), TiO2 in water (Tseng andLin, 2003), and CuO in water with temperature

change (Kulkarni et al., 2006). The problem withthese equations is that they do not reduce to theEinstein equation at very low particle volumeconcentrations and, hence, lack a sound physicalbasis.

Table 2: Models for effective viscosity

Investigator Equation

Einstein (1906) ( )eff p b1 2.5 = +

Simha (1940)( )

( )

( )

( )

2 2

eff p b

1 a / c 3 a / c1 14

15 ln 2a / c 1.5 ln 2a / c 0.5 = + + +

Simha (1940) ( )( )

eff p b

16 a / c1

15 arctan a / c = +

Eilers (1941) ( )( ){ }

eff p b

p p max2

2

p max p b

1.251

1 /

1 2.5 1.5625 2.5 /

= + =

+ + + +

De Bruijn (1942) ( )2eff b p p b2p p

11 2.5 4.698

1.25 1.552 = = + + +

+

Kuhn and Kuhn (1945)( )

( )

( )

( )

2 2

eff p b

1 a / c 3 a / c1 24

15 ln 2a / c 1.5 ln 2a / c 0.5 = + + +

Vand (1948) ( )2

eff p p b1 2.5 7.349 = + + +

Vand (1948) ( )p2.5 2eff b p p be 1 2.5 3.125

= = + + +

Robinson (1949) ( )p p 2eff b p p p r p br p

c1 1 c c s

1 s

= + = + + +

Saito (1950) ( )2eff p b p p bp

2.51 1 2.5 2.5

1 = + = + + +


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Continuation Table 2

Table 2: Models for effective viscosity

Investigator Equation

Saito (1950) ( )2eff p b p p bp

2.51 1 2.5 2.5

1 = + = + + +

Mooney (1951)( )( )

( ){ }eff p p p max b

2

p max p b

exp 2.5 / 1 /

1 2.5 3.125 2.5 /

= =

+ + + +

Brinkman (1952)( )

( )eff b p 2 b2.5p

11 2.5 4.375

1

= = + + +

Simha (1952) ( ){ }2eff p p max p b1 2.5 125 / 64 = + + +

Eshelby (1957)p

eff p b p b p

p

115 151 1 , 1/ 3

2 4 5 7

= + = +

Frankel and Acrivos (1967)( )

( )

1/ 3

p p max

eff b1/ 3

p p max

/9

81 /

=

Krieger (1972) ( )( ) ( )

eff b1.82

p p max

2 2

max p p max p b

1

1 /

1 1.82 / 2.5662 /

= =

+ + +

Lundgren (1972) ( )2eff b p p bp

11 2.5 6.25

1 2.5 = = + + +

Batchelor (1977) ( )2eff p p b1 2.5 6.2 = + +

Graham (1981)

( ) ( ) ( )eff p b2

p p p p p p

4.51 2.5

s / r 2 s / r 1 s / r

= + + + +

Phan-Thien and Graham (1991)p p max

eff b2

p p max

1 0.5( / )a1 1.461 0.138

c 1 ( / )

= + +

Liu and Masliyah (1996) ( )( ) ( )

( ){ }

2 2

eff 1 p max p 2 p max p b2

p p max

2 2

1 p 2 p max p b

1c 2 / c 6 /

1 /

1 c c 3 /

= + +

= + + +

Tseng and Lin (2003) eff p b13.47exp(35.98 ) =

Maga et al. (2004) ( )2eff p p b1 7.3 123 = + +

Maga et al. (2004) ( )2

eff p p b1 0.19 306 = +

Koo and Kleinstreuer (2005)

( ) ( )4 BBrownian b p p pp p

k T5 10 134.63 1722.3 0.4705 6.04 T

2 r = + +

,

where the particle motion related empirical parameter

0.8229

p p

0.7272

p p

0.0137(100 ) 0.01

0.0011(100 ) 0.01

< =

>

Kulkarni et al. (2006) ( ) ( )( )2 2eff p p ppln 2.8751 53.548 107.12 1078.3 15857 20587 1/ T = + + + +


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Heat Transfer Coefficient

Since the heat transfer performance is a better

indicator than the effective thermal conductivity for

nanofluids used as coolants in transportation andother industries, the modeling of nanofluid heat

transfer coefficients is gaining attention fromresearchers. However, it is still at an early stage, andthe theoretical models for nanofluid heat transfer

coefficients are quite limited. All the equations aremodified from traditional equations such as the

Dittus-Boelter equation (Dittus and Boelter, 1930) or

the Gnielinski equation (Gnielinski, 1976) with

empirical parameters added. Therefore, these

equations are only valid for certain nanofluids oversmall parameter ranges. More experimental and

theoretical studies are needed before general models

can be developed and verified.Recently, Polidori et al. (2007) investigated the

natural convection heat transfer of Newtonian Al2O3 /water nanofluids in a laminar externalboundary-layer using the integral formalism

approach. Based on a macroscopic modeling and

under the assumption of constant thermophysical

nanofluid properties, it is shown that natural

convection heat transfer is not solely characterized

by the nanofluid effective thermal conductivity andthat the sensitivity to the viscosity model used seems

undeniable and plays a key role in the heat transferbehavior.

Mansour et al. (2007) investigated the effect of

uncertainty in the physical properties of

Al2O3 /water nanofluid on its thermohydraulicperformance for both laminar and turbulent fully

developed forced convection in a tube with uniform

heat flux. Since the effects of certain nanofluid

characteristics such as average particle size andspatial distribution of nanoparticles on these

properties are not presently known precisely, it is

quite difficult to conclude as to the presumed

advantages of nanofluids over conventional heat

transfer fluids. More experimental data regardingthese effects are needed in order to assess the true

potential of nanofluids.

Table 3: Models of effective heat transfer coefficient

Investigator Nanofluids Correlation

(Pak and Cho, 1998)Al2O3-water, TiO2-water,

turbulent0.8 0.5

Nu 0.021Re Pr=

(Das et al., 2003a) Al2O3-water, pool boilingm 0.4

bNu c Re Pr= ,

c and m are particle volume concentration dependent parameters.

(Xuan and Li, 2003) CuO-water, turbulent ( )0.6886 0.001 0.9238 0.4p pNu 0.0059 1.0 7.6286 Pe Re Pr= +

(Yang et al., 2005)

graphite-in-transmission fluid,

graphite-synthetic oil mixture,laminar

( ) ( )1 / 3 0.14m 1 / 3 bNu c Re Pr D L = ,c and m are nanofluid and temperature dependent empiricalparameters.

(Buongiorno, 2006) turbulent

( )( )

( ) ( )1/ 2 2 / 3f / 8 Re 1000 Pr

Nu1 f / 8 Pr 1

+

=

+ ,

where the dimensionless thickness of the laminar sublayer + is

an empirical parameter.

(Polidori et al., 2007)Al

2O

3, Newtonian laminar,

natural convection

( )

4

r r

b2nf r nf 1/ 4

k4 5Nu Gr

3 378v 9 5

=

(UWT)

( )

4

r rb2 4

r nf nf 1/ 5

2 k6Nu Gr5 27v 9 5

=

(UWT)

Tthermal boundary layer thickness

that of the dynamical one

= =


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NUMERICAL SIMULATION

For numerical simulations two approaches have

been adopted in the literature to investigate the heat

transfer characteristics of nanofluids. The firstapproach assumes that the continuum assumption is

still valid for fluids with suspended nano-sizeparticles. The other approach uses a two-phase

model for a better description of both the fluid andthe solid phases, but it is not common in the open

literature. The single phase model is much simpler

and computationally more efficient. Anotherapproach is to adopt the Boltzmann theory. The heat

transfer enhancement using nanofluids may be

affected by several factors such as the Brownianmotion, layering at the solid/liquid interface, ballistic

phonon transport through the particles, nanoparticleclustering and the friction between the fluid and the

solid particles. It is difficult to describe all these

phenomena mathematically, however.Maga et al. (2004a,b) numerically investigated

the hydrodynamic and thermal characteristics ofnanofluids flowing through a uniformly heated tube(L = 1 m) in both laminar and turbulent regimesusing the single phase model with adjustedproperties. Results showed that the addition ofnanoparticles can increase the heat transfersubstantially compared to the base fluid alone. It wasalso found that ethylene glycol- Al2O3 providedbetter heat transfer enhancement than water- Al2O3nanofluids. However, Maga et al. (2005) alsodiscussed the disadvantages of nanofluids withrespect to heat transfer. The inclusion ofnanoparticles introduced drastic effects on the wallshear stress, which increased with an increase of thesolid volume fraction. A new correlation wasproposed by Maga et al. (2006) to describe thethermal performance of Al2O3 /water nanofluids

under turbulent regime, 0.71 0.35fullyNu 0.085Re Pr= ,

which is valid for 4 510 Re 5 10 , 6.6 Pr 13.9 and

0 10% .

Roy et al. (2004) conducted a numerical study of

heat transfer for water-Al2O3 nanofluids in a radial

cooling system. They found that addition ofnanoparticles in the base fluids increased the heat

transfer rates considerably. Use of 10 vol%

nanoparticles resulted in a two-fold increase of the

heat transfer rate as compared to that of the pure basefluid. Their results are similar to those of Maiga

et al. (2004a,b) since they both used the same model.Wang et al. (2006) investigated numerically free

convective heat transfer characteristics of a two-

dimensional cavity over a range of Grashof numbers

and solid volume fractions for various nanofluids.

Their results showed that suspended nanoparticlessignificantly increased the heat transfer rate at all

Grashof numbers. For water- Al2O3 nanofluid, theincrease of the average heat transfer coefficient was

approximately 30% for 10 vol% nanoparticles. Themaximum increase in heat transfer performance of

80% was obtained for 10 vol% Cu nanoparticlesdispersed in water. Furthermore, the average heat

transfer coefficient was seen to increase by up to

100% for a nanofluid consisting of oil containing 1vol% carbon nanotubes. Furthermore, the presence

of nanoparticles in the base fluid was found to alter

the structure of the fluid flow for horizontal

orientation of the heated wall.Khanafer et al. (2003) developed an analytical

model to determine natural convective heat transfer

in nanofluids. The nanofluid in the enclosure was

assumed to be a single phase. The effect ofsuspended nanoparticles on a buoyancy-driven heat

transfer process was analyzed. It was observed that

the heat transfer rate increased as the particle volumefraction increased at any given Grashof number. Kim

et al. (2004) analytically investigated the instabilityin natural convection of nanofluids by introducing a

new factor (f), which measures the ratio of the

Rayleigh number of the nanofluids to that of the base

fluids, to describe the effect of nanoparticle addition

on the convective instability and heat transport of the

base fluid. Results demonstrated that the increasedparticle volume fraction improves the heat transfer

rates in nanofluids compared to those in the basefluid alone.

Xuan and Roetzel (2000) derived several

correlations for convective heat transfer ofnanofluids. Both single phase and two phase models

were used to explain the mechanism of the increased

heat transfer rates. However, there are few

experimental data to validate such models. Jang andChoi (2004a) investigated the natural stability of

water-based nanofluids containing 6nm copper and2nm diamond nanoparticles in a rectangular cavity

heated from the bottom. They noted that nanofluids

were more stable compared to the base fluids.Recently, Abu-Nada (2008) numerically

investigated heat transfer over a backward facingstep (BFS) with nanofluids using the finite volume

method. They found that the average Nusselt number

increased with the volume fraction of nanoparticlesfor the whole range of Reynolds number

( 200 Re 600 ) studied. Numerical simulation ofnatural convection in horizontal concentric annuli,


14/18


15/18



out outer tube (-)pe modified nanoparticle (-)

p nanoparticle (-)

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